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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 665.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
665.a1 | 665d2 | \([0, -1, 1, -16660, -1081562]\) | \(-511416541770305536/214587319023035\) | \(-214587319023035\) | \([]\) | \(3000\) | \(1.4579\) | |
665.a2 | 665d1 | \([0, -1, 1, -210, 6798]\) | \(-1029077364736/18960396875\) | \(-18960396875\) | \([5]\) | \(600\) | \(0.65320\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 665.a have rank \(1\).
Complex multiplication
The elliptic curves in class 665.a do not have complex multiplication.Modular form 665.2.a.a
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.